I'm in the post-joining the Council of Crowns part of the game, and the scientists are apparently working on a machine using the cog and a gold-making machine. They keep asking for more money and more staff, and I was just wondering, is there actually a payoff to these plotlines? Or are they just there to get you to give away your money?

i dunno about the cog, i sold mine to the sultan, but after some time the gold making machine got finished. it gives you at least 20 gold a day, sometimes that and an aditional 60 gold, still not sure if it was worth the investment but eh

The cog quest eventually causes a chain of events of like 3-5 days where people report missing scrap metal or rabbits with a decrease happiness each time. Then the scientist comes back with a giant machine controlled by a rabbit. The machine comes back every so often and captures criminals and asking if y'all are friends. Gives you happiness stat.

Yes there IS payoff. Believe me i also wanted to stop funding them.

The gold machine, once finished, will give you either 20/60/150 gold at the end of the day. And sometimes it activates twice.

The cog will give you a rabbit mech that gives you a bit of happiness. But apparently you could've waited and sold the cog to the sultan for 1000 gold according to the comments here...